[問題] IMO 2013 in Colombia Day1

[FAlin (FA（バルシェ應援）)]

1. Prove that for any two positive integers k , n there exist positive
   integers m_1 , m_2 , ... , m_k such that

        2^k - 1          1          1             1
   1 +  ------- = ( 1 + --- )( 1 + --- )...( 1 + --- ) .
           n            m_1        m_2           m_k

2. Giver 2013 red and 2014 blue points in the plane , no three of them on a
   line. We aim to split plane by lines (not passing through these points)
   into regions such that there are no regions containing points of both the
   colors. What is the least number of lines that always suffice?

3. Let ABC be a triangle and that A_1 , B_1 , and C_1 be points of cantact of
   the excircles with the sides BC , AC , and AB , respectively. Prove that if
   the circumcenter of △A_1B_1C_1 lies on the circumcircle of △ABC , then
   △ABC is a right triangle.

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推 Dawsen:有請各位學長做難度分析~
→ FAlin:第三題沒難度!
推 cmrafsts:今年TWN1可能BRONZE
推 cmrafsts:第三題用geogebra是沒難度，但現場做會畫不準...
推 myflame:1還算簡單 寫起來有點費事就是(或者我的方法不夠好)
推 Dawsen:我想知道樓上的解法，寫起來費事的應該是妙解
推 myflame:ok 這幾天沒電腦 晚點補上
推 cuttlefish:原po不是冥燈大大嗎?xd